Question: Liam opened a savings account and deposited $\$6000$. The account earns $5\%$ in interest annually. He makes no further deposits and does not withdraw any money. In $t$ years, he has $\$8865$ in this account. Write an equation in terms of $t$ that models the situation.
Explanation: The strategy This problem involves an initial deposit of money increasing by $5\%$ every year. So, to find the amount of money in the account over time, we repeatedly multiply the original deposit, $\$6000$, by $1.05$. [Why?] Because of this, we know we can model the situation with an exponential expression of the form $a(1+r)^x$, where $a$ is $6000$ and $r$ is $0.05$. We now only need to find $x$, which represents the number of times the interest is calculated. Finding the exponent Since the interest is annual, this means it is calculated one time per year. Because $t$ represents the number of years in this case, our exponent is simply $t$. Writing an equation We can now replace $x$ in the original model with $t$. Therefore, the expression $6000\cdot(1.05)^t$ models the amount of money in the account after $t$ years. Since we know that in $t$ years, Liam has $\$8865$ in this account, we can set the above expression equal to $8865$. $8865= 6000\cdot(1.05)^{t}$ The answer An equation that models the situation is $6000\cdot (1.05)^t=8865$.